# Buoyancy

In physics, buoyancy is the upward force on an object produced by the surrounding liquid or gas in which it is fully or partially immersed, due to the pressure difference of the fluid between the top and bottom of the object. The net upward buoyancy force is equal to the magnitude of the weight of fluid displaced by the body. This force enables the object to float or at least to seem lighter. Buoyancy is important for many vehicles such as boats, ships, balloons, and airships, and plays a role in diverse natural phenomena such as sedimentation.

## Archimedes' principle

It is named after Archimedes of Syracuse, who first discovered this law. Vitruvius (De architectura IX.9–12) recounts the famous story of Archimedes making this discovery while in the bath (for which see eureka) but the actual record of Archimedes' discoveries appears in his two-volume work, On Floating Bodies. The ancient Chinese child prodigy Cao Chong also applied the principle of buoyancy in order to measure the accurate weight of an elephant, as described in the Sanguo Zhi.

This is true only as long as one can neglect the surface tension (capillarity) acting on the body.[1]

The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (specifically if the surrounding fluid is of uniform density). Thus, among objects with equal masses, the one with greater volume has greater buoyancy.

Suppose a rock's weight is measured as 10 newtons when suspended by a string in a vacuum. Suppose that when the rock is lowered by the string into water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs will be 10 newtons minus the 3 newtons of buoyant force: 10 − 3 = 7 newtons. This same principle even reduces the apparent weight of objects that have sunk completely to the sea floor, such as the sunken battleship USS Arizona at Pearl Harbor, Hawaii. It is generally easier to lift an object up through the water than it is to finally pull it out of the water.

The density of the immersed object relative to the density of the fluid is easily calculated without measuring any volumes:

${\displaystyle {\frac {\mbox{Density of Object}}{\mbox{Density of Fluid}}}={\frac {\mbox{Weight}}{{\mbox{Weight}}-{\mbox{Apparent immersed weight}}}}\,}$

## Forces and equilibrium

Pressure increases with depth below the surface of a liquid. Any object with a non-zero vertical depth will see different pressures on its top and bottom, with the pressure on the bottom being higher. This difference in pressure causes the upward buoyancy force.

The hydrostatic pressure at a depth h in a fluid is given by

${\displaystyle P=\rho hg\,}$

where

${\displaystyle \rho \,}$ is the density of the fluid,
${\displaystyle h\,}$ is the depth (negative height), and
${\displaystyle g\,}$ is the standard gravity (${\displaystyle \scriptstyle \approx \,}$ -9.8 N/kg on Earth)

The force due to pressure is simply the pressure times the area. Using a cube as an example, the pressure on the top surface (for example) is thus

${\displaystyle F_{\mathrm {top} }=d^{2}\rho h_{\mathrm {top} }g\,}$

where ${\displaystyle d}$ is the length of the cube's edges. The buoyant force is then the difference between the forces at the top and bottom

${\displaystyle F_{\mathrm {buoyancy} }=d^{2}\rho h_{\mathrm {top} }g-d^{2}\rho h_{\mathrm {bottom} }g\,}$

which reduces to

${\displaystyle F_{\mathrm {buoyancy} }=d^{2}\rho g({h_{\mathrm {top} }}-{h_{\mathrm {bottom} }})\,}$

in the case of a cube, the difference in ${\displaystyle h\,}$ between the top and bottom is ${\displaystyle -d\,}$, so

${\displaystyle F_{\mathrm {buoyancy} }=-d^{3}\rho g\,}$

or

${\displaystyle F_{\mathrm {buoyancy} }=-\rho Vg\,}$
where V is the volume of the cube, ${\displaystyle d^{3}\,}$

The negative magnitude implies that it is in the opposite direction to gravity. It can be demonstrated mathematically that this formula holds true for any submerged shape, not just a cube.

The buoyancy of an object depends, therefore, only on two factors: the object's submerged volume, and, the density of the surrounding fluid. The greater the object's volume and surrounding density of the fluid, the more buoyant force it experiences. Thus the magnitude of the buoyant force is simply equal to the weight of the displaced fluid. In this context, displacement is the term used for the weight of the displaced fluid and, thus, is an equivalent term to buoyancy.

The total force on the object is thus the net force of buoyancy and the object's weight

${\displaystyle F_{\mathrm {net} }=mg-\rho Vg\,}$

If the buoyancy of an (unrestrained and unpowered) object exceeds its weight, it tends to rise. An object whose weight exceeds its buoyancy tends to sink.

It is common to define a buoyant mass mb that represents the effective mass of the object with respect to gravity

${\displaystyle m_{b}=m_{\mathrm {o} }\cdot \left(1-{\frac {\rho _{\mathrm {f} }}{\rho _{\mathrm {o} }}}\right)\,}$

where ${\displaystyle m_{\mathrm {o} }\,}$ is the true (vacuum) mass of the object, whereas ρo and ρf are the average densities of the object and the surrounding fluid, respectively. Thus, if the two densities are equal, ρo = ρf, the object appears to be weightless. If the fluid density is greater than the average density of the object, the object floats; if less, the object sinks.

### Compressive fluids

The atmosphere's density depends upon altitude. As an airship rises in the atmosphere, its buoyancy reduces as the density of the surrounding air reduces. The density of water is essentially constant: as a submarine expels water from its buoyancy tanks (by pumping them full of air) it rises because its volume stays the same (the volume of water it displaces if it is fully submerged) while its weight is decreased.

### Compressible objects

As a floating object rises or falls the forces external to it change and, as all objects are compressible to some extent or another, so does the object's volume. Buoyancy depends on volume and so an object's buoyancy reduces if it is compressed and increases if it expands.

If an object at equilibrium has a compressibility less than that of the surrounding fluid, the object's equilibrium is stable and it remains at rest. If, however, its compressibility is greater, its equilibrium is then unstable, and it rises and expands on the slightest upward perturbation, or falls and compresses on the slightest downward perturbation.

Submarines rise and dive by filling large tanks with seawater. To dive, the tanks are opened to allow air to exhaust out the top of the tanks, while the water flows in from the bottom. Once the weight has been balanced so the overall density of the submarine is equal to the water around it, it has neutral buoyancy and will remain at that depth. Normally, precautions are taken to ensure that no air has been left in the tanks. If air were left in the tanks and the submarine were to descend even slightly, the increased pressure of the water would compress the remaining air in the tanks, reducing its volume. Since buoyancy is a function of volume, this would cause a decrease in buoyancy, and the submarine would continue to descend.

The height of a balloon tends to be stable. As a balloon rises it tends to increase in volume with reducing atmospheric pressure, but the balloon's cargo does not expand. The average density of the balloon decreases less, therefore, than that of the surrounding air. The balloon's buoyancy reduces because the weight of the displaced air is reduced. A rising balloon tends to stop rising. Similarly a sinking balloon tends to stop sinking.

## Density

If the weight of an object is less than the weight of the fluid the object would displace if it were fully submerged, then the object has an average density less than the fluid and has a buoyancy greater than its weight. If the fluid has a surface, such as water in a lake or the sea, the object will float at a level where it displaces the same weight of fluid as the weight of the object. If the object is immersed in the fluid, such as a submerged submarine or air in a balloon, it will tend to rise. If the object has exactly the same density as the fluid, then its buoyancy equals its weight. It will tend neither to sink nor float. An object with a higher average density than the fluid has less buoyancy than weight and it will sink. A ship floats because although it is made of steel, which is more dense than water, it encloses a volume of air and the resulting shape has an average density less than that of the water.

## References

1. "Floater clustering in a standing wave: Capillarity effects drive hydrophilic or hydrophobic particles to congregate at specific points on a wave" (PDF). 2005-06-23.